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Properties

Label 728.2.z.a
Level $728$
Weight $2$
Character orbit 728.z
Analytic conductor $5.813$
Analytic rank $0$
Dimension $168$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [728,2,Mod(99,728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(728, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("728.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 728.z (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.81310926715\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(84\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 12 q^{6} - 12 q^{8} + 168 q^{9} - 16 q^{16} + 12 q^{18} - 20 q^{20} + 24 q^{22} - 32 q^{24} + 44 q^{26} - 20 q^{34} - 16 q^{40} - 8 q^{41} - 52 q^{44} - 32 q^{46} - 64 q^{48} + 72 q^{50} - 56 q^{52}+ \cdots - 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1 −1.41371 − 0.0376584i −1.29962 1.99716 + 0.106476i −1.07967 + 1.07967i 1.83728 + 0.0489415i 0.707107 + 0.707107i −2.81940 − 0.225737i −1.31100 1.56700 − 1.48568i
99.2 −1.41163 + 0.0854949i 2.45227 1.98538 − 0.241374i 2.76045 − 2.76045i −3.46169 + 0.209657i 0.707107 + 0.707107i −2.78198 + 0.510470i 3.01364 −3.66073 + 4.13274i
99.3 −1.40861 − 0.125800i 0.804590 1.96835 + 0.354406i −1.31095 + 1.31095i −1.13335 − 0.101218i 0.707107 + 0.707107i −2.72805 − 0.746838i −2.35264 2.01153 − 1.68170i
99.4 −1.40544 + 0.157306i 0.322497 1.95051 − 0.442167i 0.536770 − 0.536770i −0.453249 + 0.0507306i −0.707107 − 0.707107i −2.67176 + 0.928264i −2.89600 −0.669960 + 0.838834i
99.5 −1.39814 + 0.212598i −3.11840 1.90960 − 0.594484i −1.91075 + 1.91075i 4.35997 − 0.662965i −0.707107 − 0.707107i −2.54351 + 1.23715i 6.72442 2.26528 − 3.07772i
99.6 −1.39500 + 0.232335i −0.0645954 1.89204 − 0.648214i 2.31204 − 2.31204i 0.0901105 − 0.0150078i −0.707107 − 0.707107i −2.48879 + 1.34384i −2.99583 −2.68812 + 3.76246i
99.7 −1.38171 + 0.301445i 1.78478 1.81826 − 0.833020i −2.78184 + 2.78184i −2.46605 + 0.538012i −0.707107 − 0.707107i −2.26121 + 1.69910i 0.185435 3.00514 − 4.68228i
99.8 −1.37091 − 0.347291i −1.38625 1.75878 + 0.952208i 2.25385 − 2.25385i 1.90042 + 0.481432i 0.707107 + 0.707107i −2.08043 − 1.91620i −1.07831 −3.87257 + 2.30708i
99.9 −1.33902 − 0.455004i −1.49360 1.58594 + 1.21852i −1.52676 + 1.52676i 1.99995 + 0.679592i −0.707107 − 0.707107i −1.56918 − 2.35323i −0.769169 2.73905 − 1.34968i
99.10 −1.30868 − 0.536051i 3.16260 1.42530 + 1.40304i −1.63317 + 1.63317i −4.13885 − 1.69532i 0.707107 + 0.707107i −1.11316 − 2.60017i 7.00207 3.01275 − 1.26183i
99.11 −1.28652 − 0.587244i −2.05702 1.31029 + 1.51101i 0.203896 − 0.203896i 2.64640 + 1.20797i −0.707107 − 0.707107i −0.798389 − 2.71341i 1.23132 −0.382053 + 0.142580i
99.12 −1.25686 + 0.648310i −1.56353 1.15939 − 1.62967i 0.919330 − 0.919330i 1.96514 − 1.01365i 0.707107 + 0.707107i −0.400660 + 2.79991i −0.555373 −0.559458 + 1.75148i
99.13 −1.23981 + 0.680346i 3.34691 1.07426 − 1.68700i 0.829436 − 0.829436i −4.14954 + 2.27706i −0.707107 − 0.707107i −0.184134 + 2.82243i 8.20183 −0.464040 + 1.59265i
99.14 −1.21923 + 0.716575i −0.540843 0.973040 − 1.74734i −2.54410 + 2.54410i 0.659411 − 0.387554i 0.707107 + 0.707107i 0.0657398 + 2.82766i −2.70749 1.27881 − 4.92489i
99.15 −1.21480 + 0.724064i 0.295013 0.951462 − 1.75918i 1.00711 − 1.00711i −0.358381 + 0.213609i −0.707107 − 0.707107i 0.117929 + 2.82597i −2.91297 −0.494220 + 1.95264i
99.16 −1.21122 − 0.730033i 1.01834 0.934103 + 1.76846i 0.224245 − 0.224245i −1.23343 − 0.743422i −0.707107 − 0.707107i 0.159632 − 2.82392i −1.96298 −0.435316 + 0.107903i
99.17 −1.19188 − 0.761198i 0.715638 0.841155 + 1.81451i 1.31309 − 1.31309i −0.852955 − 0.544742i 0.707107 + 0.707107i 0.378648 − 2.80297i −2.48786 −2.56457 + 0.565525i
99.18 −1.17472 + 0.787426i 2.30698 0.759920 − 1.85001i −0.308723 + 0.308723i −2.71005 + 1.81658i 0.707107 + 0.707107i 0.564051 + 2.77161i 2.32216 0.119566 − 0.605758i
99.19 −1.16349 − 0.803927i −3.07399 0.707402 + 1.87072i −1.42853 + 1.42853i 3.57655 + 2.47126i 0.707107 + 0.707107i 0.680867 − 2.74525i 6.44941 2.81051 − 0.513642i
99.20 −1.08722 − 0.904404i 2.69981 0.364106 + 1.96658i −1.71234 + 1.71234i −2.93529 − 2.44172i −0.707107 − 0.707107i 1.38272 − 2.46741i 4.28897 3.41035 − 0.313048i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.84
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
13.d odd 4 1 inner
104.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 728.2.z.a ✓ 168
8.d odd 2 1 inner 728.2.z.a ✓ 168
13.d odd 4 1 inner 728.2.z.a ✓ 168
104.m even 4 1 inner 728.2.z.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.z.a ✓ 168 1.a even 1 1 trivial
728.2.z.a ✓ 168 8.d odd 2 1 inner
728.2.z.a ✓ 168 13.d odd 4 1 inner
728.2.z.a ✓ 168 104.m even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(728, [\chi])\).